How trading volume responds to return in financial dynamics?

How trading volume responds to price return is investigated by the return–volume correlation dynamics, based on the daily data of two typical financial markets, i.e., the Chinese and the United States stock markets. Whereas the price returns being observed to be positively correlated with the trading volumes for the Chinese stock markets, a significant transition from the positive correlation to negative correlation is revealed for the United States stock markets. Leverage and anti-leverage effects of the return–volume correlations are found for the last decade of the United States stock markets and the Chinese stock markets, respectively. Nonlocal dynamics further suggests an adverse correlation for the two markets. A retarded herding model is introduced to describe the leverage and anti-leverage effects. Our work may shed a new light on the financial dynamics differing in the mature and emerging markets.